1.
Basic
A 1 kg mass hangs from a spring with stiffness k = 25 N/m. There is no damping. Find the natural angular frequency ω0.
Show answer
Answer:
Use ω0 = sqrt(k/m).
Oscillations & resonance practice problems
In this topic you solve mx'' + cx' + kx = F. The unknown x(t) is a displacement: a mass on a spring, a charge in a circuit, a swing angle. You find the natural frequency, classify the damping, and compute the response to a periodic force. Recognition cue: the equation contains x'' together with x, and the solution swings back and forth or decays. You will practice: Free undamped oscillation, Amplitude and phase form, Three damping cases, Forced oscillation and resonance.
2.
Basic
Solve x'' + 9x = 0 with x(0) = 2 and x'(0) = 0.
Show answer
Answer:
ω0 = 3. The initial values give C1 = 2 and C2 = 0.
3.
Basic
A signal is x(t) = 3 cos 2t + 4 sin 2t. Find its amplitude R.
Show answer
Answer:
Use R = sqrt(C1² + C2²).
4.
Basic
Solve x'' + 4x = 0 with x(0) = 0 and x'(0) = 6.
Show answer
Answer:
ω0 = 2. C1 = x(0) = 0 and C2 = x'(0)/ω0 = 3.
5.
Basic
A 4 kg buoy floats in water. When it sits x meters below its rest level, the water pushes it up with force 16x N, so 4x'' + 16x = 0. Find the period T of the bobbing motion.
Show answer
Answer:
Divide by 4 to get x'' + 4x = 0.
6.
Basic
Find the general solution of x'' + 6x' + 9x = 0.
Show answer
Answer:
c² − 4mk = 36 − 36 = 0, so critical damping with double root r = −3.
7.
Basic
A model car suspension obeys x'' + x' + 4x = 0. Classify the damping with one word: overdamped, underdamped, or critical.
Show answer
Answer:
Here m = 1, c = 1, k = 4.
The result is negative, so the motion is underdamped.
8.
Basic
Find the general solution of x'' + 5x' + 4x = 0.
Show answer
Answer:
Factor the characteristic equation.
Real roots −1 and −4, so overdamped.
9.
Basic
A circuit has a 1 H inductor, a 0.01 F capacitor, and no resistor. The charge obeys q'' + 100q = 0. At t = 0 the charge is q(0) = 0 and the current is q'(0) = 5. Find q(t).
Show answer
Answer:
ω0 = 10. C1 = q(0) = 0 and C2 = q'(0)/ω0 = 1/2.
10.
Basic
An eardrum model obeys x'' + 4x = 6 cos t. Find a particular solution x_p.
Show answer
Answer:
ω0² = 4 and ω² = 1 are not equal, so use the standard formula.
Check: x_p'' + 4x_p = −2 cos t + 8 cos t = 6 cos t.
Want these problems to schedule themselves? Inkerval brings handwritten practice and spaced repetition together on iPad.
Be first in when the doors open. No spam, ever.
More differential equations practice
- What an ODE is practice problems
- Separable equations practice problems
- First-order linear practice problems
- Exact & substitutions practice problems
- First-order modeling practice problems
- Second-order homogeneous practice problems
- Nonhomogeneous equations practice problems
- Laplace transforms practice problems
- Linear systems practice problems